Stochastic model reduction: convergence and applications to climate equations

Abstract

We study stochastic model reduction for evolution equations in infinite-dimensional Hilbert spaces and show the convergence to the reduced equations via abstract results of Wong–Zakai type for stochastic equations driven by a scaled Ornstein–Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling.

Singular Limit of BSDEs and Optimal Control of Two Scale Stochastic Systems in Infinite Dimensional Spaces

“…Stochastic dynamic models that evolve based on distinct and divergent time scales constitute a well-established area of research, encompassing both controlled and uncontrolled scenarios. This field has found extensive applications, as evidenced by classical results detailed in [21], and more recent applications, such as those explored in [8] and [4], particularly in the realms of neural networks and climate models.…”

The stochastic primitive equations with non-isothermal turbulent pressure

“…In the first derivation, given in Subsections 2.1-2.2, we motivate the noise leading to the non-isothermal turbulence balance (1.1e) by borrowing ideas from stochastic climate modeling (see e.g. [MTVE01,AFP21] and the reference therein). In the second one, worked out in Subsection 2.3, we derive (1.1) by looking at the Navier-Stokes equations as a two-scale system, where large and small scales are given by the horizontal and vertical ones, respectively; see Figure 1.…”

Quantitative mixing and dissipation enhancement property of Ornstein-Uhlenbeck flow

“…In this section, we prove some auxiliary results concerning the time increments of the process T solution of (1.1). Proofs are mostly inspired by our previous works [1,18].…”

“…Finally, in section 5 we prove Proposition 3.3. Its proof is based on a discretization procedure very common in the literature about averaging and Wong-Zakai approximations theorems for stochastic differential equations, and are inspired by our previous works [1,18].…”